Some people wonder why I am attmepting to solve what is a pretty complex problem without resorting to high powered computer applications and listing dozens of references. Part of the reason is I remember a comment that one climate scientist said, "Our undergraduates solve for climate sensitivity with back of the envelop calculations."
That's fine, if they can do it so can I. To me it is just an entertaining puzzle. Something to do to while away some time. So my goal is to shgow how to solve for climate sensitivity, ACCURATELY, with back of the envelop calculations.
One of the somewhat difficult calculations to do long hand is the S-B relationships. F~5.67e-8(T^4) is a little complicated. So I simplify like this;
5.67e-8 is the main constant in the Stefan-Bolzmann equation. I say main, because emissivity has to be considered and the average for space, 0.926 J/K varies as, well, the energy per degree with respect to the radiant properties of the source. One of the things we need to know is the combined emissivity of the various sources and the transmittance of the media between them.
For a CO2 doubling we can assume an object in the atmosphere is a S-B energy source. If it emits energy F at T1, then its difference in energy emitted at T2 would be,
dF/dT=[ 5.67e-8(T2)^4 – 5.67e-8(T1)^4]/(T2-T1),
Instead finishing a formal derivation, let’s just select a reasonable T2=255K and T1=254K then,
5.67e-8(255^4-254^4)/(255-254)=5.67e-8(42.3e8-41.6e8)/1=5.67*(0.7)/1, note that the e-8 and the e+8 can be added which simplifies the result to 5.67*0.7 equals 3.969= dF/dT @ dT254, or 4F/T @ 254K.
Using the same relationship, dF/dT@288K=5.44 or 5.44F/T A 288K
So it should be obvious that the change in flux F is dependent on the temperature of the body and the emissivity of the body. For a point source of CO2 forceing, if the source temperature is ~255K, 4 is a good approximation of the change in flux with respect to a small change temperature and for a point source at 288K 5.44 is a good approximation for the change in flux with respect to change in temperature.
You can simply graph this relationship on appropreate paper, interpolate between the ranges or plug the whole shebang into your computer if that blows wind up your skirt.
The main point is that the surface to TOA relationship is 5.44/4 or 1.35 and for a TOA source with respect to the surface 4/5.44 or 0.74.
The emissivity is one of the larger questions anyway, so by using the perfect black body relationships, I have a back of the envelop type equation that is accurate and very flexible. "Simplicity is elegance" as my old professor from Bell Labs was fond of saying.
Since the Arrhenius equation is based on the Stefan-Boltzmann relationship;
The doubling of CO2 will produce an increase in forcing or resistant to OLR based on the Arrhenius equation equal to Delta F=Alpha ln(Cf/Co) = Alpha * 0.69. The constant alpha is based on the Stefan-Boltzmann relationship for black body radiation. Since I am using the perfect black body approximation I will use 5.67 for alpha. Generally, 5.67 times the 0.926 or ~ 5.25 is used for alpha. Since CO2 is assigned 33% of the greenhouse forcing or .33*152=50.16, a doubling of CO2 would increase that to 55.8 which with 4.53Wm-2 per degree would be 55.8/4.53 = 12.3 degrees versus 50.16/4.53=11.07 or 1.23 degrees of warming at the surface.
How some ever, that at the surface depends on where the CO2 forcing takes place.
Most guys can remember .69 and 5.67 is now burned into my memory.